Thermal radiation heat transfer [VOL-3]

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NASA SP-164

THERMAL RADIATION HEAT TRANSFER

Volume Ill Radiation Transfer With Absorbing,

Emitting, and Scattering Media

N A T I O N A L AERONAUTICS A N D SPACE A D M I N I S T R A T I O N

NASA SP-164

THERMAL HEAT TRANSFER Volume III Radiation Transfer With Absorbing, Emitting, and Scattering Media

Robert Siegel and John R. Howell Lewis Research Center Cleveland, Ohio

Scientific and Technical I?~formationOffice 1971 NATIONAL AERONAUTICS AND SPACE ADMINISTRATION Washington, D.C.

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

Price $1.75 Library of Congress Catalog Card Number 67-62877

PREFACE This is the third and final volume of the series "Thermal Radiation Heat Transfer" that is being published as the NASA Special Publication SP-164. The first and second volumes appeared in 1968 and 1969, respectively. As stated in the Preface to volume I, this publication is an outgrowth of a course in thermal radiation that has been given as part of an internal advanced study program at the NASA Lewis Research Center. This volume contains nine chapters. The first discusses some of the fundamentals of absorption of radiation along a path in amedium, and emission by the medium. The important property of the radiation intensity is developed showing that along a path in a nonattenuating nonemitting medium the intensity is invariant with position. Thus the magnitudes of any attenuation or emission can be expressed in terms of the changes produced in the intensity with distance. In chapter 2 the absorption, scattering, and emission effects along a path are combined to develop the equation of transfer. The solution of this equation gives the intensity within the medium. The intensity can then be used to obtain energy fluxes. These are related to the local temperature within the medium by means of the energy conservation equation. This provides the necessary relations to obtain energy transfers and temperature distributions in the medium. Some methods for solving these equations are given in chapter 3. These methods include various approximate techniques, a very important one being the diffusion approximation. The boundary conditions and application of the diffusion method are considered in detail. Before any radiation solution can be applied, the radiative properties must be known. Gas radiation properties vary eonsiderably with wavelength. In chapter 4 some of the fundamentals of the radiative properties are discussed. These include line broadening mechanisms and band absorption correlations. Chapter 5 considers the important situation of radiation from a gas that is well mixed so that it is isothermal. The method of analyzing heat transfer in a gas filled enclosure is developed by extending the net radiation method presented in volume I1 for enclosures containing a nonabsorbing medium. The very useful concept of radiative mean beam length is derived for use in gas energy exchange. iii

iv

THERMAL RADIATION HEAT TRANSFER

Chapters 6, 7, and 8 provide further development by discussing the Monte Carlo method, radiation combined with conduction and convection, and scattering. Finally, chapter 9 treats a number of specialized topics such as radiation in media with nonunity refractive index, and radiation from flames.

CONTENTS CHAPTER

1

PAGE

FUNDAMENTALS O F RADIATION IN A B Sc3RBING. EMITTING. AND SCATTERING MEDIA ........................................................................ 1.1 INTRODUCTION .......................................................... 1.2 SYMBOLS .................................................................... 1.3 PHYSICAL MECHANISMS O F ABSORPTION AND EMISSION ....................................................................... 1.4 SOME FUNDAMENTAL PROPERTIES OF THE RADIATION INTENSITY ...................................................... 1.5 THE ATTENUATION OF ENERGY .................................. 1.5.1 The Extinction Coefficient ............................................ 1.5.2 Radiation Mean Penetration Distance ............................. 1.5.3 Optical Thickness ....................................................... 1.5.4 The Absorption Coefficient........................................... 1.5.5TrueAbsorptionCoefficient.......................................... 1.5.6 The Scattering of Energy ............................................. 1.6 THE EMISSION O F ENERGY ......................................... 1.7 DEFINITIONS USED FOR ENGINEERING GAS PROPERTIES ..................................................................... 1.7.1 Absorptance ............................................................. 1.7.2 Emittance ................................................................. 1.7.3 Transmittance............................................................ 1.8 THE CONCEPT O F LOCAL THERMODYNAMIC EQUILIBRIUM ....................................................................... 1.9 CONCLUDING REMARKS ............................................. REFERENCES .....................................................................

2

T H E EQUATIONS O F TRANSFER F O R AN A B SORBING-EMITTING GAS .................................... 2.1 INTRODUCTION .......................................................... 2.2 SYMBOLS .................................................................... 2.3 THE EQUATION O F TRANSFER .................................... 2.3.1 Derivation ................................................................. 2.3.2 Integration by Use of Integrating Factor .......................... 2.4 ENERGY CONSERVATION WITHIN THE MEDIUM .......... 2.4.1 Radiative Equilibrium .................................................. 2.4.2 Some Mean Absorption Coefficients ............................... 2.5 EQUATION O F TRANSFER FOR PLANE LAYER ............. 2.6 THE GRAY GAS ........................................................... 2.6.1 Transfer Equations ..................................................... 2.6.2 Plane Layer Between Black Plates ................................. 2.6.3 Use of Exponential Integral Functions ............................ 2.7 ENERGY RELATIONS BY USE O F PHOTON MODEL ......... 2.8 CONCLUDING REMARKS ............................................. REFERENCES .....................................................................

1 I 3 4 8 12 13 14 14 15 19 20 21

24 24 25 29 32 34 34

vi

THERMAL RADIATION HEAT TRANSFER PAGE

CHAPTER

3

APPROXIMATE SOLUSTIONS OF THE EQUATION OF T M N S F E R ................................................ 3.1 INTRODUCTION .......................................................... 3.2 SYMBOLS .................................................................... 3.3 APPROXIMATE SOLUTIONS BY NEGLECTING TERMS IN THE EQUATION O F TRANSFER ............................. 3.3.1 The I'ransparent b a s Approximation .............................. 3.3.2 The Emission Approxin~ation........................................ 3.3.3 The Cold Medium Approximation .................................. 3.4 DIFFUSION METHODS IN RADIATIVE TRANSFER .......... 3.4.1 Simplified Derivation of the Diffusion Equation ................ 3.4.2 The General Radiation Diffusion Equation ....................... 3.4.2.1 The Rosseland equation for local radiative flux............ 3.4.2.2 The emissive power jump as a boundary condition ......... 3.4.2.3 The emissive power jump between two absorbing-emitting regions ........................................................ 3.4.2.4 Summary............................................................. 3.4.3 Use of the Diffusion Solution ........................................ 3.4.3.1 Gray stagnant gas between parallel plates .................. 3.4.3.2 The discontinuity in emissive power between two gas reglons .............................................................. 3.4.3.3 Other diffusion solutions for gray gases ..................... 3.4.4 Final Remarks ........................................................... 3.5 APPROXIMATIONS BY USING MEAN ABSORPTION COEFFICIENTS .............................................................. 3.5.1 Some Mean Absorption Coefficients ............................... 3.5.2 Approximate Solutions of the Transfer Equations Using Mean Absorption Coefficients ..................................... 3.6 APPROXIMATE SOLUTION O F THE COMPLETE EQUATION O F TRANSFER ................................................. 3.6.1 The Astrophysical Approximations ................................. 3.6.1.1 The Schuster-Schwarzschild approximation ............... 3.6.1.2 The Milne-Eddington approximation .......................... 3.6.2 The Differential Approximation ..................................... 5.6.2.1 Houndary conditions ............................................. 3.6.2.2 Applications of the differential approximation ............. 3.7 CONCLUDING REMARKS ............................................. REFERENCES .....................................................................

4

AN INTRODUCTION T O THE MICROSCOPIC BASIS FOR RADIATION IN GASES AND GAS PROPERTIES ............................................................ 4.1 INTRODUCTION ......................................................... 4.2 SYMBOLS .................................................................. 4.3 SOME ELEMENTS O F QUANTUM THEORY .................. 4.3.1 Bohr Model of Hydrogen Atom ...................................... 4.3.2 Schriidinger Wave Equation ....................................... 4.4 INDUCED EMISSION AND THE PLANCK DISTRIBUTION .........................................................................

113 113 113 115 115 117 121

CONTENTS CHAPTER

PAGE

4.5 THE EQUATION OF TRANSFER ................................... 4.6 THE ABSORPTION PROPERTIES OF GASES ................ 4.6.1 Spectral Line Broadening ........................................ 4.6.1.1 Natural broadening ................................................ 4.6.1.2Dopplerbroadening ............................................... 4.6.1.3Collisionbroadening ............................................ 4.6.1.4Starkbroadening ............................................... 4.6.2 Absorption or Emission by a Spectral Line ...................... 4.6.3 Continuum Absorption ................................................ 4.6.3.1 Bound-free processes ............................................. 4.6.3.2 Free-free processes ............................................... 4.6.4 Band Absorption Correlations ................................. 4.7 CONCLUDING REMARKS ............................................. REFERENCES .....................................................................

5

vii

T H E ENGINEERING TREATMENT OF GAS RADIATION IN ENCLOSURES ............................. 5.1 INTRODUCTION .......................................................... SYMBOLS ................................................................. NET RADIATION METHOD FOR ENCLOSURE FILLED WITH ISOTHERMAL GAS- SPECTRAL RELATIONS ... 5.3.1 Definitions of Spectral Transmission and Absorption Factors .................................................................. 5.3.2 Matrix of Enclosure Theory Equations ........................... 5.3.3 Heat Balance on Gas ................................................... 5.4 EVALUATION O F SPECTRAL GEOMETRIC MEAN TRANSMITTANCE AND ABSORPTANCE FACTORS ...... 5.4.1 Hemisphere to Differential Area at Center of Its Base ......... 5.4.2 Top of Right Circular Cylinder to Center of Its Base ............ 5.4.3 Side of Cylinder to Center of Its Base ............................ 5.4.4 Entire Sphere to Any Element on Its Surface or to Its Entire Surface .................................................................. 5.4.5 Infinite Plate to Area on Parallel Plate ........................... 5.4.6 Rectangle to a Directly Opposed Parallel Rectangle .......... 5.5 THE MEAN BEAM LENGTH FOR RADIATION FROM AN ENTIRE GAS VOLUME TO ALL OR PART O F ITS BOUNDARY ............................................................ 5.5.1 Mean Beam Length for Gas Between Parallel Plates Radiating to Area on Plate ............................................... 5.5.2 Mean Beam Length for Sphere of Gas Radiating to Ally Area on Boundary ................................................... 5.5.3 Radiation from Entire Gas Volume to Its Entire Boundary in Limit When Gas Is Optically Thin ........................... 5.5.4 Correction for Mean Beam Length When Gas Is Not Optically Thin ............................................................ 5.6 TOTAL RADIATION EXCHANGE IN BLACK ENCLOSURE BETWEEN ENTIRE GAS VOLUME AND ENCLOSURE BOUNDARY BY USE O F MEAN BEAM LENGTH ............ 5.6.1 Radiation from Gas to All or Portion of Boundary ............... 5.6.2 Exchange Between Entire Gas Volume and Boundary ......... 5.2 5.3

125 128 128 131 131 132 134 134 137 137 138 138 153 153

157 157 157 159 164 164 166 168 168 169 171 172 173 174

175 177 177 178 179

183 183 187

...

vlll

CHAPTER

6

THERMAL RADIATION HEAT TRANSFER PAGE 5.7 TOTAL MDIATION EXCHANGE IN ENCLOSURE BY INTEGRATION O F SPECTRAL EQUATIONS ............... 188 5.7.1 Band Equations ......................................................... 189 189 5.7.2 Transmission and Absorption Factors ............................. 5.8 RADIATION THROUGH NONISOTHERMAL GASES ......... 201 5.8.1 The Curtis-Godson Approximation ................................ 202 5.8.2 The Zoning Method ..................................................... 207 211 5.8.3 The Exchange Factor Approximation .............................. REFERENCES ..................................................................... 219

T H E MONTE CARLO TECHNIQUE ....................... 6.1 INTRODUCTION .......................................................... 6.2 SYMBOLS .................................................................... 6.3 DISCUSSION OF THE METHOD ................................... 6.4 RADIATION THROUGH GRAY GASES .......................... 6.4.1 Infinite Parallel Planes ................................................ 6.4.2 Infinitely Long Concentric Cylinders .............................. 6.4.3 Radiation Between Adjacent Gray Keg~ons..................... 6.5 CONSIDERATION O F RADIATIVE PROPERTY VARIATIONS ....................................................................... 6.6 COUPLING WITH OTHER HEAT TRANSFER MODES ..... 6.7 TRANSIENT RADIATION PROBLEMS ............................ 6.8 INCORPORATION O F SCATTERING PHENOMENA......... 6.9 CONCLUDING REMARKS .............................................. REFERENCES ......................................................................

7

ENERGY T R A N S F E R BY RADIATION COMB I N E D W I T H CONDUCTION A N D / O R CONVECTION ...................................................................... 7.1 INTRODUCTION .......................................................... 7.2 SYMBOLS .................................................................... 7.3 RADIATION WITH CONDUCTION ................................ 7.3.1 The Conduction-Radiation Parameter ............................. 7.3.2 Energy Balance ........................................................... 7.3.3 Plane Layer Geometry .................................................. 7.3.4 Simple Addition of Radiation and Conduction Energy Transfers ............................................................... 7.3.5 The Diffusion Method ................................................... 7.3.6 The Exchange Factor Approximation ............................... 7.4 CONVECTION, CONDUCTION, AND RADIATION ...... 7.4.1 Boundary Layer Problems ............................................. 7.4.1.1 Optically thin thermal layer ...................................... 7.4.1.2 Optically thick thermal layer .................................... 7.4.2 Channel Flows ............................................................ 7.4.3 Other Multimode Problems ....................................... 7.5 CONCLUDING REMARKS .............................................. REFERENCES ......................................................................

221 221 221 223 229 229 231 233 234 237 237 238 238 238

241 241 241 243 244 246 247 251 254 260 264 265 266 268 270 272 272 273

ix

CONTENTS

PAGE

CHAPTER

8

RADIATIVE TRANSFER IN SCATTERING AND ABSORBING MEDIA ................................................ 8.1 INTRODUCTION .......................................................... 8.2 SYMBOLS .................................................................... 8.3 SOME IMPORTANT QUANTITIES IN THE DESCRIP-

TION O F SCATTERING ............................................. 8.3.1 The Scattering Cross Section ......................................... 8.3.2 The Phase Function ..................................................... 8.4 SCATTERING FROM VARIOUS TYPES O F PARTICLES ..... 8.4.1 A Cloud of Large Specularly Reflecting Spheres ............... 8.4.2 Reflection From a Diffuse Sphere .................................... 8.4.3 Large Dielectric Sphere with Refractive Index Close to

Unity ...................................................................... 8.4.4 Diffraction From a Large Sphere ................................... 8.4.5 Rayleigh Scattering ...................................................... 8.4.5.1 Scattering cross sections for Rayleigh scattering......... 8.4.5.2 Phase function for Rayleigh scattering ........................ 8.4.6 Mie Scattering Theory ................................................. 8.5 RADIATIVE TRANSFER IN SCATTERING MEDIA .......... 8.5.1 The Equation of Transfer in a Pure Scattering Atmosphere ... 8.5.2 Scattering in Absorbing-Emitting Media ..........................

REFERENCES .....................................................................

9

SOME SPECIALIZED EFFECTS IN ABSORBING-RADIATING MEDIA. ............................. 9.1 INTRODUCTION .......................................................... 9.2 SYMBOLS .................................................................... 9.3 RADIATION PHENOMENA IN MEDIA WITH NONUNITY

REFRACTIVE INDEX .................................................

9.3.1 Media With Constant But Nonunity Refractive Index .......... 9.3.2 The Effect of Brewster's Angle ....................................... 9.3.3 Radiative Transfer Between Dielectrics Spaced Closely

Together ................................................................. 9.4 FLAMES, LUMINOUS FLAMES, AND PARTICLE RADIA-

TION ........................................................................

9.4.1 Theoretical Flame Temperature ...................................... 9.4.2 Radiation From Nonluminous Flames .............................. 9.4.3 Radiation From and Through Luminous Flames ................. 9.4.3.1 Experimental correlation of soot spectral absorption ...... 9.4.3.2 Electromagnetic theory prediction of soot spectral

absorption .......................................................... 9.4.3.3 Total emittance of soot cloud .................................... 9.4.4 Radiation From Gases Containing Luminous Particles ......... 9.5 LUMINESCENCE .......................................................... 9.6 TRANSIENT RADIATION PROBLEMS ...........................

REFERENCES ..................................................................... APPENDIX .................................................. .. ................................... INDEX ................................................................................................

Chapter 1. Fundamentals of Radiation in Absorbing, Emitting, and Scattering Media 1.1

INTRODUCTION

The study of energy transfer through media that can absorb, emit, and scatter radiation has received increased attention in the past several years. This interest stems from the complicated and interesting phenomena associated with nuclear explosions, hypersonic shock layers, rocket propulsion, plasma generators for nuclear fusion, and ablating systems. Although some of these applications are quite recent, the study of gas radiation has been of continuing interest for over 100 years. One of the early considerations was the absorption of radiation in the Earth's atmosphere. This has always plagued astronomers when observing on Earth the light from the Sun and more distant stars. Figure 1-1 shows the observed form of the solar spectrum recorded by Samuel Langley over a period of years beginning in 1880. The dashed curve shows the estimated solar emission curve (assumed to be a blackbody spectrum at 5600 K) and the solid curve shows the spectrum after atmospheric attenuation (ref. 1). The absorption occurs in specific wavelength regions, illustrating that gas radiation properties vary considerably with wavelength. The atmospheric absorption of solar radiation is chiefly by water vapor and carbon dioxide. Extensive discussions of absorption in the atmosphere are given by Goody (ref. 2) and Kondratyev (ref. 3). Gas radiation has also been of interest to astrophysicists with regard Solar spectrum measured o n Earth

---- Solar emission spectrum

-

2.

E'

W C

I

1.1

1.4

1.8

2.6 Wavelength, A, pin

4.4

FIGURE 1-1. -Attelluatio~l of solar spectrum by Earth's atmosphere as measured by Langley (ref. I).

2

THERMAL RADIATION HEAT TRANSFER

to the study of stellar structure. Models of stellar atmospheres, such as for the Sun, and the energy transfer processes within them have been constructed; then the emitted energy spectra calculated on the basis of the models are compared with observed stellar spectra. In industry the importance of gas radiation was recognized in the 1920's in connection with heat transfer inside furnaces. The carbon dioxide and water vapor formed as products of combustion were found to be significant emitters and absorbers of radiant energy. Radiation can also be appreciable in engine combustion chambers where peak temperatures reach a few thousand degrees. The energy emitted from flames depends not only on the gaseous emission but also arises from the heated carbon (soot) particles that are formed within the flame. Another interesting example of radiation within an absorbing-emitting medium is in a glass melting furnace. As described in reference 4, the temperature distribution measured within a deep tank of molten glass was found to be more uniform than that expected from heat conduction alone. It was thought that convection might account for the discrepancy, but experimental investigations did not indicate that this was the contributing heat transfer mode. In the late 1940's it became evident that radiative transfer by absorption and reemission within the glass provided a significant means of energy transport. Two difficulties are encountered in studying radiation within absorbing, emitting, and scattering media that make these studies, to say the least, challenging. First, absorption and emission of energy are occurring not only at system boundaries, but at every local point within the medium. Scattering is also a local transfer process within the medium. A complete solution of the energy exchange problem therefore requires knowledge of the temperature and physical properties of the medium at every point within the system. The mathematics describing such a situation is inherently complex. A second difficulty is that spectral effects are often much more pronounced in gases than for solid surfaces. As a result a detailed spectrally dependent analysis may be required. When approximations are used based on spectrally averaged properties, special care must be taken. Most of the simplifications introduced in gas radiation problems are aimed at dealing with one or both of these two complexities. A brief comment on the approach to radiation in gases used herein is in order. The astrophysical approach (i.e., the equation of transfer) is used for determining the local values of the radiation intensity within the medium. As will be defined in detail in section 1.4, the intensity is concerned with energy transport along a path in a single direction. By studying the variation of intensity along a path, a good understanding is obtained of how the individual processes of absorption, emission, and scattering enter into the radiative transfer. It is the most useful method

FUNDAMENTALS IN ABSORBING MEDIA

3

in problems dealing with atmospheric absorption, stellar structure, and others where the spectral intensity at some position is often a quantity of interest. Two excellent texts (refs. 5 and 6) deal in detail with this formulation as used in astrophysics. The astrophysical approach however must be adapted for more convenient use by the engineer. The engineer is chiefly interested in energy fluxes and temperatures rather than radiation intensities. Also the astrophysical notation and nomenclature is foreign to most engineers and is often inconsistent within itself. For these reasons, although fundamental ideas are developed here on the basis of the intensity of the radiation, the change to terms of local energy flux and temperature is often made. This change aids in developing useful engineering solution methods and will also show how the engineering methods can be derived in a logical manner from the astrophysical relations.

1.2 SYMBOLS area absorption coefficient concentration of gas in mixture second constant in Planck's spectral energy distribution,

hcolk

e Fo-kT

speed of light in medium other than vacuum speed of light in vacuum energy ionization potential emissive power fraction of blackbody emissive power in spectral region

0-AT

h, i K k 111,

n

P P

Q 4

R S T V CY

Planck's constant radiation intensity extinction coefficient, a (TS Boltzmann constant extinction mean free path simple refractive index pressure partial pressure of gas in mixture energy per unit time energy flux, energy per unit area and time radius of sphere coordinate along path of radiation absolute temperature volume absorptance

+

4

THERMAL RADIATION HEAT TRANSFER

cone angle, angle from normal of area emittance wave number optical thickness (eq. (1-17)) wavelength in medium frequency density Stefan-Boltzmann constant scattering coefficient transmittance solid angle Subscripts: a

b e g

i m

P s

"I

h I/

absorbed blackbody emitted gas component i mass coefficient; mean value projected source or scatter wave number dependent wavelength dependent frequency dependent

Superscripts: I

+*

directional quantity true value, not modified by addition of induced emission dummy variable of integration

1.3 PHYSICAL MECHANISMS OF ABSORPTION AND EMISSION Although this volume will be concerned with radiation in absorbing, emitting, and scattering media in general, it will almost always be gases that are used a s examples. If the radiation properties of gases and opaque solids are compared, a difference in spectral behavior is quite evident. As shown by the plots of radiation properties in chapter 5 of reference 7 (which will be referred to from this point as vol. I), the property variations with wavelength for opaque solids are fairly smooth although in some instances the variation is somewhat irregular. Gas properties however exhibit very irregular wavelength dependencies. As a result the absorption or emission by gases is significant only in certain wavelength regions, especially at temperature levels below a few thousand degrees Kelvin.

FUNDAMENTALS I N ABSORBING MEDIA Band designation, A.

Wavelength, A, p m

Wave number, 7,c m - I

FIGURE 1-2.-Low-resolution

spectrum of absorption bands for COz gas at 830 K, 10 atm, and for path length through gas of 38.8 c n ~ .

The absorptance of a gas layer as a function of wavelength typically looks as shown for carbon dioxide (COz)in figure 1-2. The radiation emitted from a solid actually originates within the solid so the solid can be considered an absorbing and emitting medium like a gas; the physics of the radiation thus has a common basis for all media. The differences in spectra are caused by the various types of energy transitions that occur within the media. A gas has different types of transitions, a fact which leads to a less continuous spectrum than for a solid. The energy transitions that account for radiation emission and absorption will now be discussed. A radiating gas can be composed of molecules, atoms, ions, and free electrons. These particles can have various energy levels associated with them. In a molecule, for example, the atoms form a dynamic system that has certain vibrational and rotational modes. These modes have specific energy levels associated with them. A schematic diagram of the energy levels for an atom, ion, or electron is shown in figure 1-3. (The levels for a molecule are diagramed in fig. 4-5.) The zero energy level is assigned to the ground state (lowest energy bound state) with the higher bound states being at positive energy levels. The energy E I

THERMAL RADIATION H E A T TRANSFER

Free states

Bound states

(a), Bound-bound absorption (b), Bound-bound emission (c), Bound-free absorption (d), Free-bound emission (e), Free-free absorption (f), Free-free emission

FIGURE 1-3. -Schematic diagram of energy states and transitions for atom, ion, or electron.

in figure 1-3 is the ionization potential, that is, the energy required to produce ionization from the ground state. Energies above El denote that ionization has taken place and free electrons have been produced. It will be convenient to discuss the radiation process by utilizing a photon or quantum point of view. The photon is the basic unit of radiative energy. Radiative emission will consist of the release of photons of energy, and absorption will be the capture of photons by a particle. When a photon is emitted or absorbed, the energy of the emitting or absorbing particle will be correspondingly decreased or increased. Figure 1-3 is a diagram of the three types of transitions that can occur. These are bound-bound, bound-free, and free-free; they will be discussed a little further on in more detail. In addition to emission and absorption processes, it is possible for a photon to transfer part of its energy in

7

FUNDAMENTALS IN ABSORBING MEDIA

certain inelastic scattering processes. These are of minor importance in engineering radiative transfer. The magnitude of the energy transition is related to the frequency of the emitted or absorbed radiation. The energy of a photon is hv where h is Planck's constant and v is the frequency of the photon energy. For an energy transition, say from bound state E3 down to bound state E2 in figure 1-3, a photon is emitted with energy E 3-E B= hv. The frequency of the emitted energy is then v = (E3- E 2 ) / h so that a fixed frequency is associated with the transition from a specific energy level to another. Thus in the absence of any other effects, the radiation emitted will be in the form of a spectral line. Conversely in a transition between two bound states when a particle absorbs energy, the quantum nature of the process dictates that the absorption is such that the particle can only go to one of the discrete higher energy levels. Consequently, the frequency of the photon energy must have certain discrete values in order for the photon to be absorbed. For example, a particle in the ground state in figure 1-3 may absorb photons with frequencies (EZ-E,)/h, (E3 -EJh, or (E4 - El)/h and undergo a transition to a higher bound energy level. Photons with other frequencies in the range 0 < v < E,/h cannot be absorbed. When a photon is absorbed or emitted by an atom or molecule and there is no ionization or recombination of ions and electrons, the process is termed a bound-bound absorption or emission (see processes (a) and (b) in fig. 1-3). The atom or molecule moves from one quantized bound energy state to another. These states can be rotational, vibrational, or electronic in molecules, and electronic in atoms. Since the bound-bound energy changes are associated with specific energy levels, the absorption and emission coefficients will be sharply peaked functions of frequency in the form of a series of spectral lines. These lines do have a finite width from various broadening effects that will be discussed in section 4.6.1. The vibrational energy modes are always coupled with rotational modes. The -rotational spectral lines superimposed on the vibrational line give a band of closely spaced spectral lines. If these are averaged together into one continuous region, it becomes a vibration-rotation band (see section 4.6.4). Rotational transitions within a given vibrational state are associated with energies at long wavelengths, 8 to 1000 pm (see fig. 1-2 of vol. I). Vibration-rotation transitions are at infrared energies of about 1.5 to 20 pm. Electronic transitions are at short wavelengths in the visible region, 0.4 to 0.7 pm, and at portions of the ultraviolet and infrared near the visible region. At industrial temperatures the radiation is principally from vibrational and rotational transitions; at high tempera-

-

FUNDAMENTALS IN ABSORBING MEDIA

9

2.4.1 of vol I). It is as if the radiation traveling through an area within the medium originated at that area. The intensity is then deJ;ned (see fig. 1-4(a)) as the radiation energy passing through the area per unit time, per unit of the projected area andper unit solid angle. The projected area is formed by taking the area that the energy is passing through and projecting it normal to the direction of travel. The unit elemental solid angle is centered about the direction of travel and has its origin at dA. The spectral intensity is the intensity per unit small wavelength interval around a wavelength A. As stated previously, the emitted intensity from a blackbody is invariant with emission angle. Now a second invariant property of intensity will be examined. Consider radiation from a source d A , traveling in an ideal medium that is nonabsorbing, nonemitting, and nonscattering and has constant properties. Suppose that an imaginary area element ~ A isI considered at distance S 1 from dA, and that dAl is normal to S1 as shown in figure 1-4(b). From the definition of spectral intensity i i , l a s the rate of energy passing through d A l per unit projected area of d A l per unit solid angle and per unit wavelength interval, the energy from d A , passing through dA1 in the direction of SI is

where the third derivative notation d3 emphasizes that there are three differential quantities on the right side of the equation. The solid angle d w ~is equal to dA,/S: so

Suppose that d A l is now placed a distance Sn from the source along the same direction a s for the original position. The rate of energy passing through dAl in the new position is

Dividing equation (1-lb) by equation (1-2) gives

Now consider a differential source emitting energy equally in all

10

THERMAL RADIATION HEAT TRANSFER

Projection of dA into plane Radiation from direction p and w i t h i n solid angl dw that passes t h r o u g h dA

.~ e r. ~ e n d i c u l toa r intensitv.

Direction of travel of intensity

(a) Geometry for definition of intensity in medium.

(b) Intensity from source to area element. ( c ) Variation of energy flux with distance from source.

FIGURE 1-4.-Derivations

(d) Intensity of emitted uadiation.

of intensity relations.

FUNDAMENTALS IN ABSORBING MEDIA

11

directions and draw two concentric spheres around it as in figure 1-4(c). If CPQA,~ is the entire spectral energy leaving the source, then the energy flux crossing the inner sphere is d 2 Q h , , y / 4 ~ Sand : that crossing the outer sphere is d 2 Q h , , / 4 ~ S $The . ratio of the energies passing through the two elements dAl is

Substituting equation (1-4) for the left side of equation (1-3) gives the following important result:

Thus, the intensity i n a given direction i n a nonattenuating and non-

emitting medium with constant properties is independent of position along that direction. Note that these intensities are based on the solid angles subtended by the source as viewed from d A 1 as in figure 1-4(b). As S is increased, the decrease in solid angle by which d A l views the source d A , is accompanied by a comparable decrease in energy flux arriving at d l .Thus the flux per unit solid angle, used in forming the intensity, remains constant. The radiant energy passing through dAl can also be written in terms of the intensity leaving the source. Using figure 1-4(d) results in

Equating this with the energy rate passing through dAl a s given by equation (1-lb) results in I

i);,l

=

",s

(1-7)

This relation again shows the invaliance of intensity with position in a nonattenuating and nonemitting medium. The invariance of intensity when no attenuation or emission is present provides a convenient way of specifying the magnitudes of any attenuation or emission as these effects are given directly by the change of intensity with distance. By use of the foregoing intensity properties, the attenuation and emission of radiation within a medium can now be considered.

12

THERMAL RADIATION HEAT TRANSFER

1.5

THE ATTENUATION OF ENERGY

Consider spectral radiation of intensity i i impinging normally on a layer of material of thickness dS as in figure 1-5. The medium in the layer absorbs and scatters radiation. For the present it will be assumed that the layer is at low temperature so that its emitted energy is negligible. As the radiation passes through the layer, its intensity is reduced by absorption and scattering. The change in intensity has been found experimentally to depend on the magnitude of the local intensity. If a which depends on the local properties coefficient of proportionality I T;L) separated by a gas at uniform temperature Tg. Equation (5-20) applied to a two surface enclosure gives (note that F1-l=FP-2=O) for k = 1 :

167

GAS RADIATION IN ENCLOSURES

-F1- 2 7 ~ .l - z e ~ b ,p d h -F

gdA

1 - 2 6 ~ , - ~ e ~ b ,

(5-22a)

and for k = 2:

For the infinite parallel plate geometry, F1-2=F2-1=1,and from and 6 ~2 - t, = 6 ~l - 2,. For equations (5-17) and (5-18) ?A, 2 - 1 = ?A, simplicity the numerical subscripts on the ? and 6 will be omitted. Then equations (5-22a) arid (5-22b) become

1 I - E A , ~ -d q h , 1 ----r ~ €A, t

EX, 2

d q z ~=, ( e ~ b I,

- ?Aehb,

2

-G

g)dA

A ~ A ~ ,

(5-23a)

Equations (5-23a) and (5-2323) are solved simultaneously for d q ~1, and After rearrangement and using the relation ( E A = 1- ? A , this yields

d q ~ 2.,

The total energy fluxes added to surfaces 1 and 2 are, respectively, 41 =

rrn

A = O&A,

1

and

q, =

jz A=O

dqA,

The total energy added to the gas in order to maintain its specified temperature is equal to the net energy removed from the plates. Hence, per unit area of the parallel plates

When the medium between the plates does not absorb or emit radiation,

168

THERMAL RADIATION HEAT TRANSFER

then ?x= 1 and ec~uations(5-24a) and (5-2413) reduce to equation (5-10) of volume 11. With an absorbing-radiating gas present the numerical integration of equations (5-24a) and (5-24b) over all wavelengths to obtain the ql and 9.2 is difficult because of the very irregular variations of the gas absorptiol: coefficient with wavelength. The integration will be further discussed in section 5.7 by dividing the wavelength range into bands of finite width that are either absorbing or nonabsorbing. 5.4

EVALUATION OF SPECTRAL GEOMETRIC MEAN TRANSMITTANCE AND ABSORPTANCE FACTORS

To compute values from the exclzailge equations, the quantities ? and 5 or (AFT) and (AFc) must be evaluated. By use of the definitioils

given in equations (5-9) and (5-10)

llj

[ I - exp (- clxs); cos P k

~ j ~ j - k j ~- k i= l, q k

71-s"

COS

Pj

dAjclAk

It is evident that it is the double integral in equation (5-27) that must be carried out for various geometrical orientations of the surfaces Aj and At. The evaluation for some specific geometries will now be considered. 5.4.1 Hemisphere to Differential Area at Center of I t s Base As shown in figure 5-4, let Aj be the surface of a henlisphere of radius R , and dAk be a differential area at the center of the hemisphere base.

FIGURE5-4.- Hemisphere filled with isothermal gas.

169

GAS RADIATION IN ENCLOSURES

Then equation (5-27) becomes, since S = R and ,Bj=O (path R is normal to hemisphere surface),

The convenient dAj is a ring element dAj=2rrR2 sin PkdPh., and the factors involving R can be taken out of the integral as R is constant for ves the hemisphere geometry. This ,j

= dAk

exp (- ahR)

By using AjdFj-dk=dAkFdk-j and noting that Fdk-j= 1, this reduces to

?h,j-d,,.= exp (- O A R )

(5-29)

This especially simple relation will be used later in conjunction with the concept of mean beam length. This is an approximate technique wherein the radiation from an actual gas volume is replaced by that from an effective hemisphere of gas.

5.4.2 Top of Right Circular Cylinder to Center of

Its Base

This geometry is shown in figure 5-5. Since Pj= P k = P , the integral in equation (5-27) becomes for the top of the cylinder Aj radiating to the element at the center of its base dAk

A convenient change in the integral is made by noting that dAj cos PIS2 is the solid angle by which the ring dAj is viewed from dAk. By considering the intersection of this solid angle with the surface of a unit hemisphere, it is found that this solid angle is also equal to 2rr sin P dp. Making this substitution results in the integral in equation (5-30) being transformed so that

AjdFj--dE;K,j-d,,.=

dAk

I,

[exp ( - a ~ S ) ] 2cos P sin

P

dp

(5-31)

Now let ( I A S = K AThen . from figure >5 cos P= ~ / S = ~ C L A /and KA, ) d P~goes ~ . from 0 to p,,,the limits on sin p dp=- d(cos p ) = ( k a ~ / ~ f As Then equation (5-31) becomes the variable K A are K A = N A / I to ( l A

w.

THERMAL RADIATION HEAT TRANSFER

FIGURE 5-5. -Geometry for exchange from top of gas-filled cylinder to center of its base.

This can be integrated by parts two times to yield

A j d F j - d k ? ~ ,j-dk=

(aih) ' d A k

(

a , i . h \ m K,, n ~ h

The last integral is a tabulated exponential integral function so the result can be evaluated without difficulty for various values of the parameters Rlh and a k h . The integral in equation (5-32) can also be found directly in terms of the exponential integral function defined in equation (2-45) by writing

GAS RADIATION IN ENCLOSURES

By letting K A = ( C ~ A - ) / ~ integrals, they become

171

and axhlp, respectively, in the two

The integral in equation (5-32) can then be written in terms of the exponential integral function as

5.4.3 Side of Cylinder to Center of Its Base Let dAj be a ring around the wall of a cylinder as shown in figure 5-6, and note that dAj cos pj/S2 is the solid angle by which dAj is viewed

FIGURE 5-6. -Geometry for exchange from side of gas-filled cylinder to center of its base.

172

THERMAL RADIATION HEAT TRANSFER

from dAk. This solid angle is also equal to 27-r sin P s dP,,. Then equatioll (5-27) can be written for the side of the cylinder to dAk as

Aj~iFj-clk?A,j-rlk = 2dAk

I,

[exp ( - a ~ s ) ]cos

Pk

sin

PI,. (@I;

6-35)

This is of the same form as equation (5-31). Let a k S = K A ; then sin P k = R / S = R U A / K and A cos f i k dPk = d(sin P k ) =- ( R c ~ A / )K~: K A Making . these substitutions and integrating by parts as for equation (5-33) yield

nih d(RIh)'+l

(5-36a)

nil8 ( H l h )

Alternatively by use of equation (5-34)

A s for equation (5-33) (or (5-34)), this result can be readily evaluated for various values of the parameters Rllz and ahh.

5.4.4 Entire Sphere to Any Element on Its Surface or to Its Entire Surface From figure 5-7 since pk = pj let them both be simply P. Then S = 2 R cos p , and using the form in equation (5-31) gives

j-d,,.=z$ exp 4R

A ~ ~ F ~ - , ~ ; A ,

2R

s=o

( - ~ A s ) s ~ s

GAS RADIATION IN ENCLOSURES

"Ak

FIGURE 5-7.- Geometry for exchange of surface of gas-filled sphere to itself.

Integrating gives

-

AjdFj-dk'Ti,j-dk=----

2d4k [ I - (2aiR (20iR)~

+ 1 ) exp (-ZCLAR)] (5-37)

which is in terms of only the single parameter 2aiR. Equation (5-37) can be integrated over any finite area A t to give the 7~ from the entire sphere to Ak as -

AjFj-k~i,j-k

=

2Ak [1 - ( 2 a ~ R 1 ) exp ( - ~ u A R ) ] (2aiR)

+

Since Fj-k=Ah./Aj (from eq. (3-65) of vol. I I ) , ' T i , j-I

where X, is the position of the previous absorption. The process of absorptions and emissions is continued until the energy bundle reaches a black boundary. This occurs when X 2 1 or X 0, and a counter S,1 or is then increased by one unit to record the absorption at the black surface. A new bundle is emitted, and the process is repeated until all N bundles have been emitted. The dimensionless net energy flux leaving surface 1 is then found from the total bundles emitted minus those reabsorbed

THERMAL RADIATION HEAT TRANSFER

Start

C Read i n input: N, KD, k

1 Set counters Sj=O, l s j l k -

S w l ' sw2 = 0 n=O

Print out results -=

o~': Q2 = --sw2 -

o~q U4 Xo = 0 l 0.8 p m

(9-39)

Then with these spectral emittances the definition in equation (9-29) can be used to evaluate the total emittance of the flame as

Some convenient graphs for use in this procedure are given in reference 18. The hope is that the Ckz obtained in this way can be applied to "simi-

342

THERMAL RADIATION HEAT TRANSFER

lar" flames. This is a very rough approximation; there are so many variables affecting the flow and mixing in the flame that it is difficult to know when the flames will have a bimilar character. The detailed nature of flames is a continuing area of active research. 9.4.4 Roldiatinn From Gases Containing Lurnineus Particles In addition to the work on hydrocarbon luminous flames in the field of combustion, other fields involve consideration of radiation from luminous gases. A common example is the luminosity in the exhaust plume of solid fueled and some liquid fueled rockets. For a solid fuel the luminosity may be caused by particles of metal that are added to promote combustion stability. The metal particles are heated to high temperatures and may undergo oxidation, thereby becoming luminous. The presence of particles in an otherwise weakly absorbing medium can cause the mixture to be strongly absorbing. "Seeding" of a gas with particles, such as finely divided carbon, has been proposed in order to increase the gas absorption (ref. 27), or as a means of shielding a surface from incident radiation (ref. 28). These techniques have possible application in connection with advanced propulsion systems. Another use for seeding is in the direct determination of flame temperatures for a nonluminous flame by the line reversal techn.iqne. In this method, a seeding material such as a sodium or cadmium salt is introduced into an otherwise transparent flame. These materials produce a strong line in the visible spectrum because of an electronic transition. The cadmium gives a red line and the sodium a bright yellow line. A continuous source such as a tungsten lamp is placed so that it may be viewed through the seeded flame with a spectroscope. The intensity seen in the spectroscope at the line wavelength is, from the integrated equation of transfer, equation (2-lo),

If the flame is assumed isothermal and of diameter D and no attenuation occurs along the remainder of the ~ a t hbetween the continuous source and the spectroscope, equation (9-41) becomes

where

SPECIALIZED EFFECTS IN RADIATING MEDIA

In the wavelength region adjacent to the ansorbing and emitting spectral line, the flame is essentially transparent so the background radiation observed adjacent to the line is il\,co,1r.601crce.Hence, by subtractin:: ii,co,lt.so,o.ce from equation (9-42) the line intensity relative to the adjacent background is

If the flame is at a higher temperature than the continuous source, equation (9-43) shows that the line intensity in the spectroscope will b e greater than the continuous background intensity. The line will then a p p e a r as a bright line imposed upon aless bright continuous spectrum in t h e spectroscope. By increasing the temperature of the continuous source, t h e source term will ovkrride. The line then appears as a dark line on a brighter continuous spectrum. If the continuous source is a blackbody a n d its ternperature is made equal to the flame temperature, then

and equation (9-43) reduces to ii The line will then dis= ii ,,,,, appear into the continuum in the spectroscope. This is b e c a u s e the absorption by the flame and the flame emission exactly compensate. If the continuous source is a tungsten lamp, the source temperature measurement is usually made with an optical pyrometer. It is noted that in the derivation of equation (9-43) it w a s assumed that the flame is transparent except within the spectral line produced by the cadmium or sodium seeding. If soot isrin the flame, t h e soot particles absorb, emit, and scatter radiation in a continuous spectrum along the path of the incident beam. The line reversal t e c h n i q u e is of less ~ r a c t i c a lutility in this instance as it then depends o n the soot behavior. The effect of soot is analyzed in reference 29. Another instance of radiation attenuation by means of p a r t i c l e s is found in the effect of dust or so-called "grains" that are b e l i e v e d to exist in the interstellar space and cause reductions in t h e observed intensity of radiation from stars (refs. 30 and 31).

9.5

LUMINESCENCE

The phenomenon of luminescence in its various forms is a fairly common one. The name covers a broad range of mechanisms t h a t result

344

THERMAL RADIATION HEAT TRANSFER

in emission of radiant energy by the transition of electrons from an excited state to a lower energy state, where the original excitation took place by means other than thermal agitation. This is a n example of a process that is not in local thermodynamic equilibrium (section 1.8). Because the electronic transitions are between discrete energy states, the span of wavelengths over which the emission occurs is quite small. Luminescence, therefore, does not add significant energy to the spectrum of emission in engineering situations and can almost invariably be neglected in engineering calculations. However, there are some situations in which effects other than total energy transport are of interest. For that reason a very brief mention of luminescence is included here. Luminescence is categorized in various ways. A common classification is by duration of the effect. Luminescence that persists over a relatively. long l o period is called "phosphorescence," a word derived from the luminescence of white phosphorus.ll Luminescence that persists only during the influence of some external exciting agent such as an ultraviolet lamp is called "fluorescence," a name arising from the strong luminescence shown by fluorspar when so irradiated. Another categorization is by description of the excitation agent. Thus, luminescence arising from a chemical reaction such as the oxidation of white phosphorus is called "chemiluminescence"; luminescence caused by a beam of incident electrons as on a color TV screen is "cathodoluminescence"; a biochemical reaction producing luminosity, as in fireflies and some marine animals, is called "bioluminescence"; luminous emission by the presence of an electric field as in certain commercial panel lamps is "electroluminescence"; and luminescence due to photon bombardment is often called "photoluminescence." The latter effect is caused by the same mechanism that causes the laser to function. Other mechanisms can cause luminescence in materials, but descriptive terms have not yet been coined. Examples are proton bombardment, which is believed to be responsible for the luminous red patches observed on the surface of the Moon, and nuclear reactions that cause luminous emission of radiation. Because luminescence is common to materials at room temperature, it obviously cannot be predicted by the usual laws that govern thermal radiation as these would predict no visible radiation a t such temperatures. This is the origin of the term cold light for fluorescent lamp emission. Rather, the quantum mechanical properties of such luminescent materials must be examined to explain their behavior. Some detailed material on luminescence is contained in references 32 and 33. T h e

"'hfeaning in comparison with relatively short, tu speak precisely. is named for the Greek word meaning "liglit carrying."

" Phosl~horusitself

SPECIALIZED EFFECTS IN RADIATING MEDIA

345

calculation of luminescence effects is outside the scope of this work and will not be treated further here.

9.6 TRANSIENT RADIATION PROBLEMS The treatment of transient phenomena when radiation is present has been sparsely treated in the literature. Some problems dealing with the effects of nuclear weapons and some situations in astrophysics require inclusion of transient effects. The equation of transfer as derived in chapter 2 neglected changes in radiation intensity with time. The equation of transfer is written for a beam of radiation of intensity i; traveling in the S direction. As the radiation travels through the differential length from S to S dS,its intensity is increased by emission and decreased by absorption. Also during the residence of the radiation within dS, the intensity can change with time. The residence time is dt= &/c where c is the speed of propagation in the medium. Hence, the change in i; can be written as

+

By substituting for di;, the equation of transfer (2-4) becomes

1 aiL(t,S) c

at

(t,S) + d i ;as = a ~ ( tS, ) [iLb(t,S ) -i;

( t , S) ]

(9-44)

Since the conditions such as temperature within the medium a r e changing with time, the absorption coefficient is a function of t i m e a s well as position. Because the speed of light is usually very large c o m p a r e d with the other quantities in the transient term, the transient t e r m i s usually very small, and the equation of transfer reverts to the steady-state form that has been given throughout this work. In some a n a l y s e s directed towards the study of nuclear weapons (refs. 17 and 18 of c h a p t e r 6), the transient term is included. To better understand the t r a n s i e n t term, consider as a simple illustration what the radiative behavior w o u l d be if a thick uniform medium at temperature T1 instantaneousf-Y had its temperature increased to a higher uniform value T2. The m e d i u m would then be at T2 but the intensity within the medium would h a v e to change from i;, ( T I ) to i;, ( T 2 ) .During this process, the radiation w o u l d not be in equilibrium. The equation of transfer reduces to ( a s s u m i n g as an aPproximation that ax can be used in the emission term, w h i c h is an equilibrium assumption),

T H E R M A L RADIATION H E A T TRANSFER

After integrating with the condition i,(=i,(,,(T,) at t = O , the result is

The radiation relaxation time (time to change by a factor of e) for equilibrium to he reestablished is thus l / c n x ( T 2 )which is usually very short for reasonable values of ax in view of the large value of the propagation velocity c in the medium. In the preceding illustration, it was assumed that the medium temperature could be instantaneously raised so that at the beginning of the transient the radiation intensity was not in equilibrium at the black radiation value corresponding to T2. Generally the temperature change of a medium would be governed by the heat capacity of the medium, and consequently transient temperature changes would be much slower than the radiation relaxation time. Hence, when coupled with the transient energy conservation equation, which contains a heat capacity term, the unsteady term in the equation of transfer would be negligible. This is why the steady form of the equation of transfer, as derived in chapter 2, can be instantaneously applied during almost all transient heat transfer processes.

EXAMPLE9-4: A gray medium is in a slab configuration originally at a uniform temperature To. The absorption coefficient is a, and t h e slab half thickness is D. The heat capacity of the medium at constant volume is c, and its density is p. At time t=O, the slab is placed in surroundings at zero temperature. Neglecting conduction and convection, discuss the solutions for the temperature profiles for radiative cooling when a is very large and when a is very small. At the slab center which is located at x = 0, the condition of symmetry provides the relation for any time,

At time t = 0 there is the condition

T=T,;

t=O,x

SPECIALIZED EFFECTS IN RADIATING MEDIA

347

As discussed in section 2.6.2 for radiation only being included. there rvill be a temperature slip at the boundaries x=-+D, so that the perature at the boundaries will be finite rather than being equal to the zero outside temperature. If heat conduction were present, the perature slip would not exist. For a large a the diffusion approximation can be employed and equation (3-25) the heat flux in the x direction is

By conservation of energy

Combining these two equations to eliminate q gives the transient energy diffusion equation for the temperature distribution in the s l a b

Defining dimensionless variables as follows:

gives

The initial condition and the boundary condition at .xs respectively,

[email protected]

7(0, t * ) c) K

become,

=O

At the boundary K = ccD, a slip condition must be used. Usir+g equation (3-45), when the surroundings are empty space at zero t e m p e r a t u r e , the et,,0=0, and the e l c = l , SO that at the exposed boundary of medium for any time

THERMAL RADIATION HEAT TRANSFER

Similar relations apply at x=- D. For these conditions solution by numerical techniques is probably necessary. For a small absorption coefficient, and since there are no enclosing radiating boundaries present, the emission approximation (section 3.3.2) can be applied. For very small a the medium is optically so thin that it is at uniform temperature throughout its thickness at any instant. From the results of example 3-2, the heat flux emerging from each boundary of the layer is q=

4aaT4D

then becomes

dT dt

pc, -=-

4aaT4

or, in dinlensionless terms,

Dimensionless time, t*

0

.2

.4

Optical depth,

.6 K / K ( X=

.8

1.0

D)

FIGCJRE9-12.-Dimensionless tenlperature profiles as a function of time for radiative cooling of a pray slab; optical thickness K ( Z = D )= 1.0. (Fro111 ref. 34.)

SPECIALIZED EFFECTS IN RADIATING MEDIA I n t e g r a t i n g w i t h t h e c o n d i t i o n t h a t O = 1 a t t"= a t u r e t h r o u g h o u t t h e slab is t h e n g i v e n by

0, t h e t r a n s i e n t t e m p e r -

(ref. 34) h a v e obtained numerical s o l u t i o l l s t o the of the complete e q u a t i o n of t r a n s f e r , a l o n g with t h e limiting s o l u t i o n s derived here. Some of t h e i r r e s u l t s for intermediate o p t i c a l thickness are shown in figure 9-12. Numerical s o l u t i o n s f o r spherical g e o m e t r i e s are found in references 35 and 36. S o l u t i o n s to m o r e i n v o l v e d t r a n s i e n t problems by use of M o n t e C a r l o are mentioned in s e c t i o n 6.7. Viskanta

and

349

Bathla

t r a n s i e n t form

REFERENCES 1. HODGMAN, CHARLESD., ED.: Handbook of Chemistry and Physics. 38th ed., Chemical Rubber Publ. Co., 1956-1957. 2. GARDON, ROBERT:The Emissivity of Transparent Materials. J. Am. Cer. Sot., vol. 39, no. 8, Aug. 1956, pp. 278-287. 3. GARDON, ROBERT:Calculation of Temperature Distributions in Glass Plates Undergoing Heat-Treatment. J. Am. Cer. Soc., vol. 41, no. 6, June 1958, pp. 200-209. 4. GARDON,ROBERT: Appendix to Calculation of Temperature Distributions ill Glass Plates Undergoing Heat Treatment (ref. 3). Mellon Inst., Pittsburgh, 1958. 5. GARDON,ROBERT:'AReview of Radiant Heat Transfer in Glass. J. Am. Cer. Sot., vol. 44, no. 7,July 1961, pp. 305-312. 6. CONDON, EDWARD U.: Radiative Transport in Hot Glass. J. Quant. S p e c t r o s c Radiat. Transfer, vol. 8, no. 1, 1968, pp. 369-385. 7. MCCONNELL, DUDLEY G.: Radiant Energy Transport in Cryogenic Condensates. Paper presented at Enpineering Colloquium, Kentucky University, Lexington, K y . , Nov. 30, 1967. 8 . CRAVAI~HO, E. G.; TIEN,C . L.; AND CAREN,R. P.: Effect of Small Spacings or1 Radiative Transfer Between Two Dielectrics. J. Heat Transfer, vol. 89, no. it, Nov. 1967, pP. 351-358. 9. ECHIGO,R.; NISHIWAKI,N.; AND HIRATA,M.: A Study on the Radiation o f Luminous Flames. Eleventh Symposium (International) On Combustion. The ~ o n l b u s t i o l l Institute, 1967, pp. 381-389. 10. PERRY,ROBERTH.; CHILTON, CECILH.; AND KIRKPATRICK, SIDNEY D., EDS. : Chemical Engineers' Handbook. Fourth ed., McGraw-Hill Book Co., Inc., 1963. 11. HOUGEN,OLAFA,; WATSON,KENNETHM.; AND RAGATZ,ROLANDA,: ater rial and Energy Balance. Vol. 1 of Chemical Process Principles. Second ed., J o h n Wiley & Sons, Inc., 1954. 12. GAYDON,A. G.; AND WOLFHARD,.H. G.: Fla~iles,Their Structure, Radiation, and Temperature. Second ed., Mac~iiillanand Co., 1960. 13. BARNETT.HENRYC.; AND HIBBARD.ROBERTR.. EDS.: Basic Considerations in the Combustion of Hydrocarbon Fuels with Air. NACA Rep. 1300,1957. 14. HOTTEL,H. C.; AND SAROFIM,A. F.: The Effect of Gas Flow Patterns o m Radiative Transfer in Cylindrical Furnaces. Int. J. Heat Mass Transfer, vol. 8, no. 8, Aug. 1965, pp. 1153-1169.

350

THERMAL RADIATION HEAT TRANSFER

15. STULL.V. ROBERT:A N D P I ~ A S SGILBERT . N.: Emissivity of Dispersed Carbon Particles. J . Opt. Soc. Arn., vol. 50, no. 2, Feb. 1960, pp. 121-129. 16. SIDDALL,R. G.; ANI) ~ I C G R A T H1., A,: The Eilrissiviiy of Luminous Flames. Ninth Symposium (International) on C:ombustion. W. G. Berl, rtl., Academic l'ress, 1963, pp. 102-110. 17. HOWARTH,C. R.; FOSTER.P. J.; AVD THRINC;.M. W.: Ttie Effect of Teniperature 011 the Extinction of Radiation by Soot Particles. Tliircl International Heat Transfer Confereilce. Vol. 5, AIChE, 1966, pp. 122-128. 18. HOTTEL,H. C.: Radiant Heat Transmission. Heat Transmission. Thirtl ed., \William H. McAdams, 5IcGraw-Hill Book Co., Inc., 1954, cli. 4. 19. HAWKSLEY,P. G. W.: The bIetliods of Particle Size A~Ieasurenrents.Part 11. Optical Illethods and Liglit Scattering. Brit. Coal Utilization Res. Assoc. Monthly Bull., vol. 16, nos. 4 and 5,1952, pp. 134-209. 20. DALZELL,W. H.; AND SAROFIM.A. F.: 0j)tical Constants of Soot and Their Application to Heat-Flux Calculations. J. Heat Transfer, vol. 91, No. I , Feb. 1969, pp. 100-104. 21. ERICKSON,W. D.; WILLIAMS,G. C.; AND HOTTEL, H. C.: Liglit Scattering Measurements on Soot in a Benzene-Air Flame. Combt~stionand Flame, vol. 8, no. 2, June 1964, pp. 127-132. 22. THRING,M. W.; BEER,J. YI.; AND FOSTER,P. J.: The Radiative Properties of 1,uminous Flanies. Third International Heat Transfer Conference. Vol. 5, AIChE, 1966, p P . 101-111. 23. THRING,M. W.; FOSTER,P. J.; MCCRATH,I. A.; AND ASHTON,J . S.: Prediction of t h e Emissivity of Hydrocarbon Flames. International Developments in Heat T r a n s f e r ASME, 1963, pp. 796-803. 24. SATO,TAKASHI;AND MATSIJMOTO,KYUICHI:Radiant Heat Transfer from L u m i n o ~ l s Flame. International Developments in Heat Transfer. ASME, 1963, pp. 804-81125. YAM, S.; A N D INOUE,H.: Radiation from Soot Particles in 1,uminous Flames. E i g h t h Sytnposiuni (International) On Combustion. Williams and Wilkins, 1962, pp. 288-293. 26. BONE, WILLIAMA.; AND TOWNEND, DONAI.DT. A.: Flame antl Comhustioll in G a s e s Lo~igmans,Green and Co., Ltd., 1927. 27. IANZO,CHESTERD.: AND RA(;SDAI.E.ROBERT(;.: Heat Transfer to a Sreded F l o w i n % (;as from an Arc Enclosed by a Quartz Tube. Proceedings of the 1964 Heat T r a n s f e r and Fluid A31echanics Institute. Warren H. Giedt and Salornon Levy, eds., S t a l l f o r d Univ. Press, 1964, pp. 226-244. 28. HOWELL,JOHN R.; AND RENKEL,HAROLDE.: Analysis of the Effect of a S e e d e d Propellant Layer on Thermal Kadiation in the Nozzlr of a Gaseous-Core N u c l e a r Propulsion System. NASA T N D-3119, 1965. 29. THOMAS,D. LYDDON:Problen~s in Applying the Line Reversal Metlrod of T e n ? . perature Measurement to Flanies. Combustion and Flame, vol. 12, no. 6, Dec. 1 9 6 8 , pp. 541-549. 30. CREENBERG, J. MAYO;A N D ROARK,T. P., EDS.: Interstellar Grains. NASA S P - ~ ~ O > 1967. 31. DONN,B.; A N D KRISHNA SWAMY,K. S.: Extinction by Interstellar Grains, Mie I'articles and Polycyclic Aronratic Molecc~les.Physica, vol. 41, no. 1, Feb. 21,1960, pp. 144-1 5 O . 32. CIJRIE,DANIEI,(G. F. J. (;.ARLICK. TRANS.):1,uminescence in Crystals. John W i l e y & Soils, Inc., 1963. PETER: Fluorescence and Phosl>llorescence. Interscience Publ., 194.') 33. PRINGSHEIM. a 34. VISKANTA,RAYILIONII; A N D BATHLA,PRITAM S.: Unsteady Energy Transfer i n Layer of Gray (;as I)y Thermal Radiation. Zeit. f. r1t1y.e~.Math. Pltys., vol. 18. n o - 3. 1967. pp. 353-367.

SPECIALIZED EFFECTS IN RADIATING MEDIA

35 1

35. VISKANTA, R.: AND LAI.L, 1'. S.: Transient Cooling of a Splrerical Mass of Hig11Temperature Gas by T h e r n ~ a lRadiation. J. Api~1.S'Iech., vol. 32, no. 4, Dec. 1965, pp. 740-746. 36. VISKANTA,R.; AND I.ALL, 1'. S.: Transient Heating a ~ l t lCooling of a Spherical Mass of Gray Gas hy Thermal Radiation. Proceedings of the Heat and Transfer Fluid blechanics Institute. 91. A. Saad and .I. 4. Sliller, eils., Stanford Univ. l'rrss. 1966, pp. 181-197.

Appendix RADIATION CONSTANTS

For the convenience of the reader, some of the important constants used in radiative transfer theory are gathered here in table I.

First Bohr electron radius Speed of light in vacuum Electron charge Planck's constant Boltzmann constant Electron mass Classical electron radius Atomic unit cross section Titomson cross section Electron volt Temperature associated with 1 eV Rest energy of electron Ionization potential of hydrogen atom

a,= h"m,e" 0.5292 X cm c,=2.9979 X 10"' cmlsec e=4.803 X 10-I" esu h = 6.625 X lo-" (erg)(sec) h = h/27r= 1.054 X (erg)(sec) k = 1.3804 X lO-'%rg/K m e = 9.108 X r , = e2/m,c; = 2.818 X 10-':I ern r a p = 0.880 X 10-1" cni2 o.r= 87rri/3 = 6.652 X 10-2%cm" 1 eV = 1.602 X 10-'"erg 1 ei7/k=11605K n ~ , c= z 8.186 X lo-' erg e2/2a,= 2.rr'e4m,,/h2= 13.60 cV

EXPONENTIAL INTEGRAL RELATIONS

A summary of some useful exponential integral relations is presented here. Additional relations are given in references 1 to 3. For positive real arguments, the nth exponential integral is defined as

and only positive integral values of n will be considered here. An alternate form is

By differentiating equation (Al) under the integral sign, t h e recurrence relation is obtained 353

THERMAL RADIATION HEAT TRANSFER

Another recurrence relation obtained by integration is n

nEll+l( x )= exp ( - x ) -xE,,(x)

2

1

(A41

Also integration results in

By use of equation (A4), all exponential integrals can be reduced to the first exponential integral given by

di

El ( x )=

p-I exp

( 5 l d p

Alternate forms of El ( x ) are

=Ix

~,(x)

til exp ( - x t ) d t =

:1

1-1

exp ( - t ) d t

For x=O the exponential integrals are equal to

For large values of x there is the asymptotic expansion

Series expansions are of the forill

(A71

APPENDIX

where y=0.577216 is Euler's constant. The general series expansion given in reference.3 is

where

and

Tabulations of E , , ( x ) are given in references 2 and 3. An abridged listing given in table I1 is included here for convenience.

THERMAL RADIATION HEAT TRANSFER

TABLE11. -VALUES OF EXPONENTIAL INTEGRALS E,,(x) [From ref. 21

E,(X)

Es(x)

1.0000 .9497 .9131 ,8817 .8535 .8278 .8040 .7818 .7610 .7412 ,7225 ,5742 ,4691 .3894 .3266 .2762 .2349 ,2009 .1724 .I485 .lo35 ,0731 .0522 .0375 .0272 .0198 .0145 .0106 .0078 .0058

0.5000 .4903 .4810 .4720 ,4633 .4549 .4468 .4388 .4311 .4236 .4163 ,3519 .3000 .2573 ,2216 .I916 .I661 .I443 .I257 ,1097 .0786 .0567 ,0412 .0301 .0221 '0163 .0120 .0089 .0066 .0049

REFERENCES 1960.

1. 2.

CHANDRASEKHAR, SUBRAHMANYAN: Radiative Transfer. Dover Publications, Inc.,

3.

ABRAMOWITZ, MILTON; AND STEGUN,IRENE A., EDS.: Handbook of ~ a t h e r n a f i ~ ~ l

KOURGANOFF, VLADIMIR:Basic Methods in Transfer Problems. Dover ~ u b l i c a t i o ~ ~ +

Inc., 1963. Functions with Formulas, Graphs, and Mathematical Tables. Appl. Math. Ser. 55$ Nat. Bur. Standards, 1964.

Index

Absorptance, 24 Absorption, 4 band correlations, 138, 146 band overlap, 152, 187 bound-bound, 7, 128 bound-free, 8, 137 free-free, 8, 137, 138 line, 7, 16, 128, 134 Absorption coefficient, definition, 13, 15 effective mean, 96 incident mean, 46,93 mass, 13 Planck mean, 45, 66, 67, 93, 224 Rosseland mean, 78,93 true, 19 Absorption factors, geometric mean, 164 geometrical, 164 Addition of radiation and conduction, 251 Angular frequency, 113 Attenuation, 12 atmospheric, 1 Bouguer's law, 13, 16 Band absorptance, 138 Band models, Elsasser, 143 exponential wide, 148 statistical, 144 Band width, correlations, 138, 148 effective, 142 limits, 196 tables, 146, 148, 149 Bohr model of atom, 115 Boltzmann distribution, 123, 132 Bouguer, Pierre, 13 Bouguer's law, 13, 16 Boundary layer with radiation, 265 optically thin layer, 266 optically thick layer, 268 Bremsstrablung, 8 Brewster's angle, 317

Broadening, line, collision, 129, 132 Doppler, 129, 131 natural, 129, 131 Stark, 129, 134 Carbon dioxide radiation, 2 band, 5,149 charts, 184, 185 mixture with water vapor, 186 Carbon monoxide band radiation, 149 Channel flows with radiation, 270 Closely spaced dielectrics, 322 Coefficient, absorption, 13, 15, 19 emission, 23 extinction, 12 scattering, 13, 20, 283 Cold medium approximation, 62, 70 Collision broadening, 129, 132 Conduction-radiation parameter, 244 Coupled problems, 241, 264 additive solution, 251 boundary layer, 265 channel flows, 270 diffusion method, 254 Monte Carlo, 237 radiation and conduction, 237, 241, 243, 246,247,251 radiation, conduction and convection, 237,241,264 Cross section, absorption, 335 scatter, 280 Curtis-Godson approximation, 202 d e Broglie, Louis, 117 Degenerate states, 121 Detailed balancing, 123 Differential approximation, 102 boundary conditions, 106 equation of transfer, 106 solutions for simple geometries, table, 109 Diffraction from sphere, 292

358

THERMAL RADIATION HEAT TRANSFER

Diffusion method, 62, 71,74, 254 jump between two absorbing-emitting regions, 82, 86 jump boundary condition, 79 radiation and conduction, 254 Rosseland diffusion equation, 74, 75, 78 scattering, 308 Diffusion solutions, concentric cylinders, 89 concentric spheres, 89, 90 parallel plates, 83 table, 89 Direct exchange area, gas-gas, 211 gas-surface, 208 surface-gas, 210 surface-surface, 209 Doppler broadening, 129, 131 Effective line width, 135 Effective band width, 142 Efficiency factor, 284 Einstein coefficients, 122 Elsasser model, 143 Emission, 4 , 2 1 from volume, 23 induced, 19,121,122 line. 134 medium with nonunity refractive index, 315 spontaneous, 19,122 Emission approximation, 62,67 Emission coefficient, 23 Emittance of gases, carbon dioxide, 5,149,184 definition, 25,145 water vapor, 149,185 Energy conservation, 43,166,246 Energy density, 56 Energy levels, 5,128 Enclosure theory, 159 band equations, 189 matrix of equations, 164 Equation of transfer, 37,38,125 approximations to, table, 62 differential form, 40 integral form, 41 plane layer, 46,49 photon model, 56, 125 Equilibrium local thermodynamic, 32 radiative, 45 Exchange factor approximation, 211, 218, 260

Exponential integrals, definition, 56 in equation of transfer, 55, 248 relations between, 353 table of values, 356 Exponential wide band model, 148, 149 Extinction coefficient, 12 Flames, 324 luminous, 331 nonluminous, 329 theoretical temperature, 325 Flux vector, 58 Gaunt factor, 128 Geometric mean beam length, 191, 192, 194 Gray gas, 83, 95,229 definition, 47 transfer equations, 48 Half width of line, 130 Induced emission, 19, 121 Intensity, 8 definition in medium, 9, 57 invariance along path in vacuum, 11 Ionization potential, 6, 117 Jump boundary condition, 72, 79, 82, 86, 88 Kirchhoff's law, 26 Krakatoa, 295 Langley, Samuel P., 1 Laser, 33 Line, broadening, 128 shape parameter, 129 strong, 137 weak, 136 width, 130, 132, 133, 135 Line reversal technique, 342 Local thermodynamic equilibrium, 32 Lorentz profile, 131 Luminous flames, 331 Luminous particles, 342 Luminescence, 343 Mean absorption coefficient, incident, 46, 93 Planck, 45,66,67,93,224 Rosseland, 78,93 Mean beam length, 175 gas not optically thin, 176, 179 geomteric, 191, 192, 194 optically thin limit, 178 table of values, 181, 192 Methane band radiation, 149

359

INDEX Mie scattering, 278, 299 Milne-Eddington approximation, 100 Monte Carlo, 221, 223 adjacent gray regions, 233 concentric cylinders, 231 parallel plates, 225, 229 Natural broadening, 129, 131 Net radiation method, 159 Nonluminous flames, 329 Optical thickness, 15, 40, 47, 3W,307 Optically thin limit, 53,67,178 Oscillator strength, 127 Parallel plates, 166, 196, 199 black with gray gas between, 49 diffusion solution, 83 Monte Carlo solution, 225,229, 231 radiation and conduction, 247 Particle radiation, 342 Penetration distance, 14 Phase function for scattering, 284 Photon, 6, 56 momentum, 117 Planck distribution, 121 Planck mean absorption coefficient, 45, 66,67,93,224 Polari~abilit~, definition, 296 table for types of scattering, 296 Population inversion, 34 Radiative equilibrium, 45 Rayleigh scattering, 294 cross section, 295 phase function, 298 Refractive index, nonunity, 315 Rosseland diffusion equation, 74, 75, 78 Rosseland mean absorption coefficient, 78,93 Rydberg constant, 117 Scattering, 238, 277 anisotropic, 20 elastic, 20, 277 equation of transfer, 302, 306 inelastic, 7, 20, 277 isotropic, 20 Scattering coefficient, 13, 20, 283 Scattering cross section, 280

Scattering from particles, diffraction from sphere, 292 large dielectric sphere with refractive index near unity, 292 large diffuse sphere, 289 large specular sphere, 286 Mie scattering, 278, 299 Rayleigh scattering, 294 Scattering optical thickness, 304 Schuster-Schwarzschild approximation, 97 Schrodinger wave equation, 117 Seeding with particles, 342 Slip, 53, 72 coefficient, 255 Soot, absorption coefficient, 333 concentration, 340 electron~agnetictheory predictions, 335 optical properties, 336 total emittance, 338 Source function, 307 Spheres, concentric, 89, 109 Spontaneous emission, 19,122 Stark broadening, 129, 134 Statistical weight, 121 Stimulated emission, 19 Strong line, 137 Theoretical flame temperature, 325 Thermodynamic equilibrium, local, 32 Transfer equation, 38,40, 125, 302,306 Transient problems, 237,246,345 Transmittance, 29 Transmittance factors, geometric mean, 164, 168 geometrical, 164 Transparent gas approximation, 62,64 True absorption coefficient, 19,40,127 Uniform gas, 15, 159,205 Vibration-rotation band, 7 Water vapor radiation, band radiation, 149 charts, 185, 186 mixture with CO?, 186 Wave function, 118 Weak line, 136 Wien distribution, 125 Zoning method, 207

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